When evaluating the \(p\) -th layer for a branch, which is the same as evaluating the \(p\) -th layer for a subvector \(f\) , it holds that \(g(x_{\text{lb}|f}) \le g(x_{\text{ub}|f})\) .
Consider now the scenario where \(x_{\text{ub}|f}=x_\text{opt}\)  and \(g(x_\text{opt}) = g(x_{\text{lb}|f})\) .
In this case, using the mentioned pruning condition, the optimal branch is considered as not valid if \(g(x_\text{opt}) \ge (1-\gamma ) \check{g}\) . This results in the algorithm returning \(\check{x}\)  when \(g(x_\text{opt})\le \check{g}\le \frac{g(x_\text{opt})}{1-\gamma }\) . This means that the algorithm allows for any solution in the \(\epsilon \) -suboptimal set given by \(\mathcal {X}_{\text{opt},\epsilon }=\left\lbrace x: \ \text{g}(x) \le g(x_\text{opt})+ \epsilon \right\rbrace \) , cf. [1]}, where \(\epsilon =g(x_\text{opt})\frac{\gamma }{1-\gamma }\) .
